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Name ______Signature ______

Econ 641 W10

Exercise 4 (corr. 1/29 as indicated in bold)

1/26, 1/28, due 2/1

1. (35 HW points) Using workfile SCHOOLS.WF1, rescale Y as well as E to be in $1000s rather than dollars, and re-estimate the equation

S = b0 + bE E + bY Y + u

a) What is the Unrestricted Sum of Squared Residuals SSRUR? ______

b) Now test the joint hypothesis H0: bE = 0, bY = 0 in this regression by re-estimating it under this restriction and comparing the SSRs. To impose these restrictions, simply eliminate E and Y from your regression, so that you are just regressing S on a constant (C) and nothing else.

What is the Restricted Sum of Squared Residuals SSRR? ______.

c) Compute the F statistic for this hypothesis from the change in SSR:

d) How many (num, den) DOF does your F-stat have? ______, ______

e) Using Table G.3b in the text, what upper and lower bounds can you place on the 5% critical value for this F-stat? ______, ______

f) Can you reject this hypothesis at the 5% level?


2. (30 HW points) Rerun the unrestricted regression,

S = b0 + bE E + bY Y + u

so that it is active in your equation window, again with both E and Y in $1000s.

a) Generate a 95% confidence ellipse for bE and bY by clicking on VIEW > COEFFICIENT TESTS > CONFIDENCE ELLIPSE in your equation window. In the first window, enter C(2), C(3) to indicate you only want the pair bE (“C(2)” in EViews) and bY (“C(3)” in EViews). Leave .95 in the second window, and leave LINE selected for INDIVIDUAL INTERVALS. Using FREEZE > TITLE, add your name at the top, print it out, and attach it to this exercise.

b) The two vertical lines are a 95% confidence interval for C(2) = bE. Can you reject the single hypothesis C(2) = 1.5 at the 5% level? ______

c) The two horizontal lines are a 95% confidence interval for C(3) = bY. Can you reject the single hypothesis C(3) = 0.8 at the 5% level? ______

d) Using your confidence ellipse, can you reject the joint hypothesis C(2) = 1.5, C(3) = 0.8 at the 5% level? ______Why? ______

e) To obtain an exact p-value for this joint hypothesis, select VIEW > COEFFICIENT TESTS > WALD-COEFFICIENT RESTRICTIONS, and enter C(2)=1.5, C(3)=0.8 in the window. Make sure the two restrictions are separated by a comma, and that each one has no interior spaces. What is the F-statistic? ______

What is its p-value? ______

Can you reject this hypothesis at the 1% test size? ______

3. (25 HW points) Use the R2 of the unrestricted regression to compute the Regression F Statistic for this equation:

How does the hypothesis tested by this Regression F statistic differ from that tested in Problem 1?


4. (20 HW points) , , … , … is an infinite sequence of random variables with the indicated distribution. In each case, find plim(), or indicate that the plim does not exist:

a) ~ N(3, 7)

b) ~ N(3, 7/n)

c) ~ N(3 + 1/n, 7n)

d) ~ N(3 + 1/n, 7/n)

5 (20 HW points) , … , , … is a sequence of estimators of a parameter q computed from samples of size 1, …, n, … . In each case, indicate whether the estimator is biased or unbiased, and if biased what the bias is:

a) ~ N(q, 7)

b) ~ N(q, 7/n)

c) ~ N(q + 1/n, 7n)

d) ~ N(q + 1/n, 7/n)


6. (20 HW points) In each of the following cases (which are the same as Problem 5), indicate whether is consistent or inconsistent, and explain briefly:

a) ~ N(q, 7)

b) ~ N(q, 7/n)

c) ~ N(q + 1/n, 7n)

d) ~ N(q + 1/n, 7/n)

7. (15 HW points) a) Using Table G.3b in the text, find the 5% critical value for the F distribution for (3, 20) (num, den) degrees of freedom ______

b) For (3, 120) (num, den) DOF ______

c) For (3, ¥) (num, den) DOF ______

d) Using Table G.4 in the text, find the 5% critical value for the c2 distribution for 3 DOF:

______

e) Divide your answer in d) by 3: ______. How does this compare to a), b) and c)?

8. (10 HW points)

If a variable Y is regressed on each of 10,000 variables Xj , none of which in fact has any explanatory power for Y, how many of the individual t-statistics would you expect to exceed the 1% critical value? (Show calculation):

9. (10 HW points)

If a variable Y is regressed on each of 10,000 variables Xj , what is the Bonferroni Adjusted test size a* for a 1% final test size a?